Second-order prestress stability and third-order rigidity of polyhedral surfaces

Daniel Robertz; Zeyuan He

REVIEW 2 major objections 4 minor 30 references

Second-order prestress stability and third-order rigidity of polyhedral surfaces

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper shows that a shaky polyhedral surface can be certified second-order prestress stable by three explicit linear-algebra conditions on a selfstress, and that no fractional rigidity orders appear before third order.

desk verdict Useful computational extension of the energy-based rigidity framework; the fractional-order proofs are genuinely sketched, but the stress-test counterexample does not land. read the letter →

arxiv 2506.22019 v2 pith:Y4PR5L4U submitted 2025-06-27 math.MG

Zeyuan He , Daniel Robertz This is my paper · ORCID

classification math.MG MSC 52C25

keywords higher-orderrigiditysecond-orderprestressstabilitypolyhedralsurfacesrigidorigamienergyfunctionalselfstressj-activetrajectoriesfoldinganglemodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Higher-order rigidity asks how many time-derivatives of a perturbation must vanish before a structure that looks rigid actually admits motion, and the classical answer misclassifies genuinely flexible mechanisms such as the double-Watt linkage. This paper adopts the recently proposed energy-based definition of higher-order rigidity — the rigidity order is half the growth rate of an associated energy along the slowest trajectory — and turns it into computable criteria for polyhedral surfaces modelled as rigid panels connected by hinges. Its central result is a test for second-order prestress stability: a selfstress stabilizes a barely-rigid first-order-flexible surface, making the prestressed energy grow quartically, if the stress matrix $\Omega = \omega \cdot d^2f/d\rho^2$ and the second-order stress tensor $\Omega_{II} = \omega \cdot d^4f/d\rho^4$ satisfy three algebraic conditions. The paper also proves that no fractional rigidity orders occur before third order and no fractional prestress-stability orders before second order, so the integer-order decision hierarchy is exhaustive over that range, and it gives a recursive linear-algebra procedure that determines the rigidity order when the rigidity matrix has nullity one. A worked planar example runs the full decision test, showing the classification in action.

What carries the argument

The carrying mechanism is the stress-tensor pair assembled from the closure constraint $f$ and a selfstress $\omega$: the stress matrix $\Omega = \omega \cdot d^2f/d\rho^2$, the second-order stress tensor $\Omega_{II} = \omega \cdot d^4f/d\rho^4$, and the cubic contraction $\omega \cdot d^3f/d\rho^3$. A $j$-active trajectory is a perturbation whose first nonvanishing derivative is of order $j$; expanding the energy along such a trajectory, every term is a contraction of these tensors, so rigidity questions reduce to quadratic-form inequalities. The decisive identity is the fourth energy derivative along a $(1,2)$ flex, $d^4E/dt^4 = 3\Omega(\rho'' \otimes \rho'') + \Omega_{II}(\rho')^4$ plus a nonnegative material-stiffness term that vanishes exactly on flexes, so its sign over the space of $(1,2)$ flexes controls stabilizability. A second load-bearing tool is the inherited equivalence between a $(j,k)$ flex and a $(j,2k)$ energy flex, which forces energy growth at orders $2j$, $4j$, $6j$ and thereby yields the no-fractional-order conclusions.

What would settle it

Take a second-order flexible framework whose rigidity matrix has nullity two and compute the Taylor expansion of the energy along every analytic perturbation through order seven: the no-fractional-order claim predicts the first nonzero energy term can only be of order $4j$, so a trajectory whose leading term is $t^5$ (rigidity order $5/2$, strictly between second and third) would refute Proposition 3, and a trajectory with leading term $t^3$ would refute the prestress-stability claim of Proposition 7.

Watch

Extended reading notes

Core claim

Working in the folding-angle model, the paper assembles the closure constraint $f$ from products of rotations around creases and writes its derivatives of every order explicitly in terms of crease direction vectors and vertex coordinates. On this basis it generalizes the energy-based definition to all $j$-active trajectories — perturbations whose first nonvanishing time-derivative has order $j$ — and shows that energy derivatives along any such trajectory are governed by the same stress matrix $\Omega = \omega \cdot d^2f/d\rho^2$ and second-order stress tensor $\Omega_{II} = \omega \cdot d^4f/d\rho^4$. The central criterion (Proposition 9) states that a second-order flexible configuration is second-order prestress stable if a selfstress $\omega$ exists such that: $\mathrm{Null}(df/d\rho)$ is $\Omega$-positive modulo $\mathrm{Null}(\Omega)$, meaning every flex direction either has positive $\Omega$-quadratic form or is annihilated by $\Omega$; $\omega \cdot d^3f/d\rho^3 = 0$; and $3\Omega(\rho'' \otimes \rho'') + \Omega_{II}(\rho')^4 > 0$ over the space of $(1,2)$ flexes. For planar surfaces the cubic condition holds automatically, because odd-order derivative contractions with the selfstress vanish. The paper further proves the absence of fractional rigidity orders below third order, and that when the rigidity matrix has nullity one the rigidity order and all flexes are computed step by step through successive rank analyses, since the indeterminacies of earlier flexes never affect the decision.

Load-bearing premise

The proofs that no fractional rigidity orders arise between second and third order, and no fractional prestress-stability orders between first and second order, extend from the first nontrivial trajectory to all faster-starting trajectories through the sentence 'Now we extend this argument for $j > 1$' rather than a written induction, so the exhaustiveness of the integer-order decision chart rests on that unshown step.

Editorial extensions

If this is right

When the rigidity matrix has nullity one, the rigidity order of a near-mechanism is decided by a sequence of rank checks on augmented matrices, and the flex at each order is computed by a Moore-Penrose solve plus one free parameter along the null direction.
Second-order prestress stability becomes a direct algebraic filter: assemble $\Omega$ and $\Omega_{II}$ from crease directions and test the three conditions of Proposition 9, with the cubic condition $\omega \cdot d^3f/d\rho^3 = 0$ satisfied automatically for planar polyhedral surfaces.
Because no fractional orders appear before third-order rigidity and second-order prestress stability, the integer-order decision flowchart is exhaustive up to those levels, and the only unresolved outcomes are the flagged 'indeterminate' cases.
The classical double-Watt paradox is explained by the difference between testing a $(1,3)$ flex and testing $(j,3j)$ flexes for all $j$: the additional coupled terms are exactly why the classical third-order criterion can declare a flexible mechanism rigid.
Although the computations are carried out in the folding-angle model, the local rigidity analysis is written for general constraint forms, so the same tests transfer to panel-hinge and point-panel models of polyhedral surfaces and to broader geometric constraint systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The pattern behind condition $\omega \cdot d^3f/d\rho^3 = 0$ suggests a general hierarchy: prestress stability at energy order $2k$ would likely require vanishing of all odd-derivative contractions $\omega \cdot d^{(2m+1)}f/d\rho^{(2m+1)}$ up to order $2k-1$, and the paper's note that higher-order stress tensors are deferred to a future article points exactly there.
The saddle-point criterion could serve as a cheap design pre-filter: any fold pattern whose null space fails to be $\Omega$-positive modulo $\mathrm{Null}(\Omega)$ for every selfstress cannot be stabilized by prestress at all, so it can be discarded before the more expensive Proposition 9 test is run.
A computational stress test of the no-fractional-order claim is feasible on small frameworks: symbolic Puiseux expansion of the energy along random analytic trajectories should never produce a leading term of order $t^5$ at a second-order flexible configuration; running such a search on frameworks with nullity two or three would be a direct way to probe the unshown inductive step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a competent, genuinely useful adaptation of the Gortler–Holmes-Cerfon–Theran energy framework to polyhedral surfaces. The new material—Prop 6's saddle-point test, Prop 9's second-order prestress stability criterion, and the recursive nullity-one flex construction in Prop 4 plus Appendix B—is concrete and directly applicable. The Taylor expansions are written out carefully, the example is worked in detail with Mathematica checks, and the reliance on Gortler et al. (2025a) and Connelly–Whiteley is explicit and fair. This is not a reorganization of the field, but it gives practitioners explicit linear-algebra tests for near-mechanisms.

The real soft spot is exactly what the reader flagged: the "no fractional orders" theorems (Propositions 3 and 7) are not fully proved. The text says "Now we extend this argument for j>1" and then jumps to the conclusion, with no complete induction. That leaves the advertised exhaustiveness of the flowchart unsupported. Also, the third-order rigidity test is only algorithmic when the rigidity matrix has nullity one; that restriction appears in Section 6 but not in the abstract, so the abstract overpromises. These are fixable with more exposition, but they are genuine gaps as written.

I checked the stress-test note and I do not think it holds. The counterexample there has df=0 and d^2f(e1,e1)=0, so the constraint set locally contains a line; the configuration is actually flexible, and flexible configurations are outside Proposition 9's domain (second-order flexible means rigid, not flexible). The broader version of the concern is also backwards: a first-order flex with Ω(v,v)>0 makes the energy grow like r², which is faster than r⁴, so it does not threaten always-4-quickly; it only helps. The dangerous directions are those with Ω(v,v)=0 but Ωv≠0, and condition [1] already rules those out. I therefore do not see a load-bearing flaw in Prop 9's sufficiency from that angle.

Who is this for? Researchers working on higher-order rigidity of frameworks, origami, and near-mechanisms. They will get clear, implementable criteria and a worked example. The paper deserves a serious referee, with the request to complete the fractional-order induction (or explicitly state it as a conjecture) and to move the nullity-one restriction into the abstract.

Referee Report

2 major / 4 minor

Summary. The paper applies an energy-based definition of higher-order rigidity, recently introduced by Gortler, Holmes-Cerfon, and Theran, to polyhedral surfaces modeled in the folding-angle formalism. It derives explicit higher-order derivatives of the closure constraints, revisits first- and second-order rigidity in terms of j-active trajectories, proposes no-fractional-order theorems for rigidity and prestress stability, gives a criterion for "second-order prestress stability" (Proposition 9), discusses the special nullity-one case for third-order rigidity, and works through a planar example. The main advertised contributions are a linear-algebra test for quartic-order stabilization by prestress and an integer-order flowchart for local rigidity.

Significance. The paper extends a recent energy-based framework to a class of geometric constraint systems of direct interest to rigid origami and deployable structures. The derivative formulas in Section 3 and Appendix A are explicit and potentially useful, the worked example in Section 8 is detailed, and the discussion of the nullity-one case gives a concrete recursive procedure. However, the central new criterion, Proposition 9, is not valid as stated: it can certify quartic energy growth along extendable flexes while missing first-order flex directions that force quadratic energy growth. Because this criterion is the paper's main theoretical and practical claim, the significance of the paper is contingent on a substantial correction.

major comments (2)

[Section 7, Proposition 9] The sufficiency claim is false as stated. Condition [1] permits a first-order flex v in Null(df/drho) with Omega(v,v) > 0 and Omega v != 0, while Condition [3] is tested only on extendable (1,2)-flexes. For any such non-extendable v, the prestressed energy along gamma(t) = rho + t v satisfies d^2 E/dt^2 at 0 = Omega(v,v) > 0, because the material-stiffness term vanishes when df . v = 0. Hence E grows like c r^2 and cannot grow tightly at order s = 4. Concretely, take two constraints on R^3 with df = 0, d^2 f_1 = diag(0,1,0), d^2 f_2 = diag(0,0,1), d^3 f = 0, and choose selfstress omega = (1,1), so Omega = diag(0,1,1). Then the (1,2)-flexes are exactly v = alpha e_1, and Conditions [1] and [2] hold; Condition [3] also holds for a suitable Omega_II with Omega_II(e_1^{otimes 4}) > 0. But e_2 is a non-extendable first-order flex with Omega(e_2,e_2) = 1, forcing order-2 growth. Proposition 9 needs at least the additional hypothesis that Null(df/drho) is contained in Null(Omega), or equivalently Condition [3] must be verified on every first-order flex direction, not only on extendable ones.
[Section 6, Proposition 3 and Section 7, Proposition 7] The no-fractional-order claims are not proved for general j > 1. In both propositions the argument is carried out for j = 1 and then extended with the sentence "Now we extend this argument for j > 1", but no induction over j is supplied. The derivative formulas for d^{ij}E/dt^{ij} contain sums over partitions whose index structure changes with j, so the j = 1 obstruction analysis does not automatically carry over. In Proposition 3, the proof constructs one family of (j,3j-1)-flexes rather than proving that every j-active second-order flex has this form. Since the flowchart in Figure 3 asserts exhaustiveness of the integer-order tests, this missing induction is load-bearing and should either be supplied or the claims should be weakened.

minor comments (4)

[Section 4 heading] The word "engrgy" in the heading should be "energy".
[Section 2] The notation R\2pi is used without a definition; it should be defined explicitly, for example as R/2pi Z.
[Section 7, Proposition 7 proof] The identity d^2E/drho^2 . (rho'' tensor rho') = 0 is asserted without stating the representative trajectory choices; for arbitrary rho'' it is not true, so the proof should specify which higher derivatives are set to zero in the j-active first-order flex.
[Section 8] The example's selfstress computation assumes that every selfstress has only the third component at each vertex or cycle; Proposition 10 should state the nondegeneracy assumption that the crease directions at each planar vertex span the plane, since otherwise in-plane selfstress components need not vanish.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; criteria are derived from the energy expansion rather than assumed, and the cited external results are load-bearing but independent.

full rationale

I walked the derivation chain from Definition 5 through Propositions 1-10. The criteria are obtained by Taylor-expanding the constraint and energy via the Faà di Bruno formula; Proposition 9 gives sufficient conditions derived from the fourth-order energy derivative, not a restatement of Definition 9. No fitted parameter is renamed as a prediction, and no input quantity is defined in terms of the claimed output. The self-citation to He and Guest (2022) supplies explicit derivative formulas with stated assumptions that do not include the target rigidity criteria, so it is independent support rather than load-bearing circularity. The cited Gortler et al. (2025a) and Connelly-Whiteley (1996) results are external to this paper. I also examined the passages that assert limitations: the inductive extension 'Now we extend this argument for j>1' in Section 7 and the analogous unstated full induction in Proposition 3 are incomplete proof details, and the sufficiency claim in Proposition 9 does not explicitly treat non-extendable first-order flexes with positive quadratic energy growth; these are correctness/completeness risks, not instances of circular derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted constants. It relies on the energy-based definition of rigidity, a cited theorem from Gortler et al., the closure-constraint map, the nullity-one restriction for the third-order test, and a planar selfstress characterization. The main mathematical object introduced, the second-order stress tensor OmegaII, is a definitional tool rather than a new physical entity.

assumptions (6)

domain assumption The energy Ei is analytic, each unstressed Ei attains a strict local minimum at fi=0, d2E/df2 is positive definite, and |omega| is small relative to the smallest material-stiffness eigenvalue (Definition 2, conditions [1]-[3]).
This is the energy-based definition of rigidity the entire paper builds on, imported from Gortler et al. (2025a) and Connelly-Whiteley (1996).
standard math Theorem 1 of Gortler et al.: for unstressed energy, a (j,k)-flex is equivalent to a (j,2k+1)-E-flex and a (j,2k)-E-flex.
Used without proof to convert kinematic flexes into energy-growth statements in Propositions 1-3 and Section 7.
domain assumption The closure constraint f assembled from Equations (1) and (2), with three components per vertex and six per representative cycle, is the local constraint map; f=0 characterizes configurations in a neighborhood.
The paper notes itself that f is only a sufficient local condition; global sufficiency is not needed.
ad hoc to paper Nullity(df/drho)=1 for the algorithmic third-order rigidity test; the general nullity case is not covered.
The only fully general recursive procedure (Prop. 4, Appendix B) assumes one-dimensional null space; the abstract omits this restriction.
domain assumption For a planar polyhedral surface, the only selfstress contributions are out-of-plane components per vertex and per cycle (Prop. 10).
Used to prove omega*di f/drho i = 0 for odd i and to set omega=[0,0,*] in the Section 8 example; the justification is a symbolic-induction sketch.
standard math Third and higher derivatives of f follow the explicit formulas from the appendix of He and Guest (2022).
A parameter-free prior derivation by one of the present authors; its stated assumptions do not include the target result, so it counts as independent support.

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Cite this review

Pith. "Pith review of Second-order prestress stability and third-order rigidity of polyhedral surfaces." pith.science (2026). https://pith.science/paper/Y4PR5L4U

@misc{pith2026250622019,
  author       = {Pith},
  title        = {Pith review of: Second-order prestress stability and third-order rigidity of polyhedral surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4PR5L4U}},
  note         = {Machine review of arXiv:2506.22019}
}

read the original abstract

There has been a longstanding confusion on the proper definition of higher order rigidity and flexibility in geometric constraint systems. Recently, an energy-based formulation of higher-order rigidity was introduced by Steven Gortler, Miranda Holmes-Cerfon, and Louis Theran (2025). In this article, we apply the framework to polyhedral surfaces and introduce new criteria for testing second-order prestress stability and third-order rigidity. Furthermore, we present comprehensive case studies of polyhedral surfaces exhibiting different levels of higher-order shakiness. These results advance the understanding of higher-order rigidity and flexibility in origami-inspired structures, which are also applicable to a broad class of near-mechanisms. Clarifying the transition from higher-order flexibility to finite mechanisms opens new directions for both theoretical investigation and mechanism design.

Figures

Figures reproduced from arXiv: 2506.22019 by the authors.

**Figure 1.** Three common modellings for polyhedral surfaces from the geometric rigidity and origami research community. The underlying graph for each model provides an abstract depiction of how bodies are connected with the geometric constraints. Here black dots represent bodies, while edges — both standard edges connecting two bodies and hyperedges connecting multiple bodies — are shown as lines and circles. A cyclic hyperedg… view at source ↗

**Figure 2.** Graphical explanation on the measurement of ‘angular and distance misfits’. (a), (b), and (c) illustrate the rotation and possible translation of local coordinate systems around three polyhedral surfaces: a degree-5 vertex, a degree-5 hole, and a representative cycle on a 6 × 4 toroidal polyhedral surface. The local coordinate systems for panels Pj−1 and Pj are depicted in red and blue, respectively, while the repre… view at source ↗

**Figure 3.** A flowchart illustrating the tests for local rigidity. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: A planar polyhedral surface which is rigid, second-order flexible and a saddle point of any prestressed energy. We use this example to show the feasibility of the local rigidity tests we proposed in [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

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Works this paper leans on

30 extracted references · 29 canonical work pages

[1]

Sufficient conditions for the extendibility of an n-th order flex of polyhedra

Victor Alexandrov. Sufficient conditions for the extendibility of an n-th order flex of polyhedra. Beitr. Algebra Geom, 39 0 (2): 0 367--378, 1998

work page 1998
[3]

Local extrema of analytic functions

Angelo Barone-Netto, Gianluca Gorni, and Gaetano Zampieri. Local extrema of analytic functions. Nonlinear Differential Equations and Applications NoDEA, 3 0 (3): 0 287--303, September 1996

work page 1996
[2]

Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks

Victor Alexandrov. Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks . Monatshefte für Mathematik, 132 0 (4): 0 269--288, April 2001

work page 2001
[4]

C. Chen. The order of local mobility of mechanisms. Mechanism and Machine Theory, 46 0 (9): 0 1251--1264, September 2011

work page 2011
[5]

Higher-order rigidity -- What is the proper definition? Discrete & Computational Geometry, 11 0 (2): 0 193--200, 1994

Robert Connelly and Herman Servatius. Higher-order rigidity -- What is the proper definition? Discrete & Computational Geometry, 11 0 (2): 0 193--200, 1994

work page 1994
[6]

Second-order rigidity and prestress stability for tensegrity frameworks

Robert Connelly and Walter Whiteley. Second-order rigidity and prestress stability for tensegrity frameworks. SIAM Journal on Discrete Mathematics, 9 0 (3): 0 453--491, 1996

work page 1996
[7]

Statics of frameworks and motions of panel structures: a projective geometric introduction

Henry Crapo and Walter Whiteley. Statics of frameworks and motions of panel structures: a projective geometric introduction. Structural Topology, 1982, núm. 6, 1982. Publisher: Université du Québec à Montréal

work page 1982
[8]

Demaine, Martin L

Erik D. Demaine, Martin L. Demaine, David A. Huffman, Thomas C. Hull, Duks Koschitz, and Tomohiro Tachi. Zero- Area Reciprocal Diagram of Origami . In Proceedings of IASS Annual Symposia , volume 2016, pages 1--10. International Association for Shell and Spatial Structures (IASS), 2016

work page 2016

Show all 30 references

[9]

N. V. Efimov. Some theorems about rigidity and non-bendability (in Russian ). Usp. Mat. Nauk, 7 0 (5): 0 215--224, 1952

1952
[10]

Gortler, Miranda Holmes-Cerfon, and Louis Theran

Steven J. Gortler, Miranda Holmes-Cerfon, and Louis Theran. Higher Order Rigidity and Energy. arXiv:2506.03108, June 2025

2025 arXiv
[11]

Zs Gáspár and T. Tarnai. Finite mechanisms have no higher-order rigidity. ACTA TECHNICA-ACADEMIAE SCIENTIARUM HUNGARICAE, 106: 0 119--126, 1994. Publisher: AKADEMIAI KIADO

1994
[12]

Kentaro Hayakawa, Zeyuan He, and Simon D. Guest. Panel-point model for rigidity and flexibility analysis of rigid origami. Computational Geometry, 121: 0 102100, August 2024

2024
[13]

Zeyuan He and Simon D. Guest. On rigid origami III : local rigidity analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 478 0 (2258): 0 20210589, February 2022

2022
[14]

The generic rank of body–bar-and-hinge frameworks

Bill Jackson and Tibor Jordán. The generic rank of body–bar-and-hinge frameworks. European Journal of Combinatorics, 31 0 (2): 0 574--588, February 2010

2010
[15]

A Proof of the Molecular Conjecture

Naoki Katoh and Shin-ichi Tanigawa. A Proof of the Molecular Conjecture . Discrete & Computational Geometry, 45 0 (4): 0 647--700, June 2011

2011
[16]

E. N. Kuznetsov. Underconstrained Structural Systems . Springer Science & Business Media, December 2012

2012
[17]

Engineering Origami : A Comprehensive Review of Recent Applications , Design Methods , and Tools

Marco Meloni, Jianguo Cai, Qian Zhang, Daniel Sang-Hoon Lee, Meng Li, Ruijun Ma, Teo Emilov Parashkevov, and Jian Feng. Engineering Origami : A Comprehensive Review of Recent Applications , Design Methods , and Tools . Advanced Science, 8 0 (13): 0 2000636, 2021

2021
[18]

Pratapa, Ke Liu, Biruta Kresling, Yan Chen, Chiara Daraio, and Glaucio H

Diego Misseroni, Phanisri P. Pratapa, Ke Liu, Biruta Kresling, Yan Chen, Chiara Daraio, and Glaucio H. Paulino. Origami engineering. Nature Reviews Methods Primers, 4 0 (1): 0 1--19, June 2024

2024
[19]

A global approach for the redefinition of higher-order flexibility and rigidity

Georg Nawratil. A global approach for the redefinition of higher-order flexibility and rigidity. Mechanism and Machine Theory, 205, 105853, March 2025

2025
[20]

On flexes associated with higher-order flexible bar-joint frameworks

Georg Nawratil. On flexes associated with higher-order flexible bar-joint frameworks. arXiv:2502.01124, February 2025

2025 arXiv
[21]

Verbiegungen höherer Ordnung und ebene Flächenrinnen

Eduard Rembs. Verbiegungen höherer Ordnung und ebene Flächenrinnen . Mathematische Zeitschrift, 36 0 (1): 0 110--121, December 1933

1933
[22]

I. Kh. Sabitov. Local Theory of Bendings of Surfaces . In Yu. D. Burago and V. A. Zalgaller, editors, Geometry III : Theory of Surfaces , pages 179--250. Springer, Berlin, Heidelberg, 1992

1992
[23]

How to recognize the order of infinitesimal mechanisms: A numerical approach

Ginevra Salerno. How to recognize the order of infinitesimal mechanisms: A numerical approach. International Journal for Numerical Methods in Engineering, 35 0 (7): 0 1351--1395, 1992

1992
[24]

Rigidity of body-bar-hinge frameworks

Király Csaba and Shin-ichi Tanigawa. Rigidity of body-bar-hinge frameworks. Handbook of Geometric Constraint Systems Principles, 435-460, 2018

2018
[25]

A proposal for a proper definition of higher-order rigidity

Hellmuth Stachel. A proposal for a proper definition of higher-order rigidity. In Tensegrity Workshop , 2007

2007
[26]

Design of infinitesimally and finitely flexible origami based on reciprocal figures

Tomohiro Tachi. Design of infinitesimally and finitely flexible origami based on reciprocal figures. Journal of Geometry and Graphics, 16 0 (2): 0 223--234, 2012

2012
[27]

Rigid folding of periodic origami tessellations

Tomohiro Tachi. Rigid folding of periodic origami tessellations. Origami, 6: 0 97--108, 2015

2015
[28]

A Proper Definition of Higher Order Rigidity

Tomohiro Tachi. A Proper Definition of Higher Order Rigidity . arXiv:2410.15541, October 2024

2024 arXiv
[29]

T. Tarnai. Higher-order infinitesimal mechanisms. Acta Technica Acad. Sci. Hung, 102 0 (3-4): 0 363--378, 1989

1989
[30]

Rigidity of molecular structures: generic and geometric analysis

Walter Whiteley. Rigidity of molecular structures: generic and geometric analysis. In Rigidity theory and applications, pages 21--46. Springer, 2002

2002

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Sufficient conditions for the extendibility of an n-th order flex of polyhedra

Victor Alexandrov. Sufficient conditions for the extendibility of an n-th order flex of polyhedra. Beitr. Algebra Geom, 39 0 (2): 0 367--378, 1998

work page 1998

[2] [3]

Local extrema of analytic functions

Angelo Barone-Netto, Gianluca Gorni, and Gaetano Zampieri. Local extrema of analytic functions. Nonlinear Differential Equations and Applications NoDEA, 3 0 (3): 0 287--303, September 1996

work page 1996

[3] [2]

Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks

Victor Alexandrov. Implicit Function Theorem for Systems of Polynomial Equations with Vanishing Jacobian and Its Application to Flexible Polyhedra and Frameworks . Monatshefte für Mathematik, 132 0 (4): 0 269--288, April 2001

work page 2001

[4] [4]

C. Chen. The order of local mobility of mechanisms. Mechanism and Machine Theory, 46 0 (9): 0 1251--1264, September 2011

work page 2011

[5] [5]

Higher-order rigidity -- What is the proper definition? Discrete & Computational Geometry, 11 0 (2): 0 193--200, 1994

Robert Connelly and Herman Servatius. Higher-order rigidity -- What is the proper definition? Discrete & Computational Geometry, 11 0 (2): 0 193--200, 1994

work page 1994

[6] [6]

Second-order rigidity and prestress stability for tensegrity frameworks

Robert Connelly and Walter Whiteley. Second-order rigidity and prestress stability for tensegrity frameworks. SIAM Journal on Discrete Mathematics, 9 0 (3): 0 453--491, 1996

work page 1996

[7] [7]

Statics of frameworks and motions of panel structures: a projective geometric introduction

Henry Crapo and Walter Whiteley. Statics of frameworks and motions of panel structures: a projective geometric introduction. Structural Topology, 1982, núm. 6, 1982. Publisher: Université du Québec à Montréal

work page 1982

[8] [8]

Demaine, Martin L

Erik D. Demaine, Martin L. Demaine, David A. Huffman, Thomas C. Hull, Duks Koschitz, and Tomohiro Tachi. Zero- Area Reciprocal Diagram of Origami . In Proceedings of IASS Annual Symposia , volume 2016, pages 1--10. International Association for Shell and Spatial Structures (IASS), 2016

work page 2016